HERE’S THE PART that everyone forgets: when Schrödinger made up the cat thing, he meant it as a complaint. Like Einstein with his aphorism about God and dice, Schrödinger thought the quantum theory he’d helped invent was goofy and incomplete, not deep. Math might require nuclei to be simultaneously stable and decayed, but cats, Schrödinger knew, aren’t at once alive and dead. But that part’s lost. When we see Schrödinger’s cat on a T-shirt, it has transitioned from complaint to icon, the sort of icon that encourages us to fix the problems of physical theory about as much as Che Guevara memorabilia moves us to gather guns for the revolution.

To be more precise, I should write that almost everyone has forgotten about Schrödinger’s cat — almost, because there’s a subgenre of popular science books that concerns itself exclusively with such “old news.” I recently read three additions to this group, and this is a report.

But lest you start snoring, let me assure you that while each of these books takes as its subject the metaphysical underpinnings of physics (yaaaaaaaawn), and while each hits all the power chords of the genre (start with the low entropy of the universe, genuflect toward Gödel, end with some shrugging over consciousness), they don’t have as much in common with each other as you might guess. There’s a range of approaches, here — a range, to be sure, that only spans from science writer to science writer — but it’s enough. Three versions of the same pop song can impart three different meanings. With these three authors we’re given three different universes.

And since that gives me some options, I may as well go in order — in publication order, but also (coincidentally, and in my own opinion) from most crazed to most rational. The possibility that this puts them in descending order of reading fun hints, if nothing else, at the pitfalls of science writing. I also think it might tell us about the human need for prophets, and about why it’s so easy to trip when you search for truth… but let’s not get ahead of ourselves.


Exhibit A: The Believer

Last year, the hubristic title was one thing that kept me away from Our Mathematical Universe: My Quest for the Ultimate Nature of Reality. Also, other reviews had made Max Tegmark sound quite insane. When I finally caved, I was surprised to find him pleasant and chatty company as he dished about his Berkeley classmates and the early universe with equal ease.

It wasn’t until page 123 that the other shoe dropped:

Hold on!!! Did I just go bananas??? I mean, so far in this book, I’ve mostly written about stuff that I hope you found pretty reasonable. Sure, some of the scientific discoveries I wrote about were controversial at the time, but at least they’re accepted by the scientific mainstream today. But then things started going kind of crazy in this chapter.

And how! What had finally spurred me to read Tegmark was the simple fact that multiple editors asked me how pieces I’d pitched to them related to his ideas. I’d always replied that the main difference between Tegmark and me is that I am not insane.

But now I know that Tegmark isn’t insane, either. Merely “bananas.”

Which I suppose could be even more dangerous.

I should admit upfront that Tegmark, a Massachusetts Institute of Technology professor and founder of the Foundational Questions Institute, has written exactly the kind of book that many readers want. Unlike either of the other books discussed here, Our Mathematical Universe has hundreds of reviewers; the top one calls it “one of the finest popular science books I’ve ever read.” I should also admit that Tegmark is a considerate and forthright writer. The book’s introduction labels chapters as “Mainstream,” “Controversial,” and “Extremely Controversial,” and it suggests which chapters the “Hard-core reader of popular science” and “Physicist” will want to skip. And when Tegmark presents the possibility of the multiverse — of multiple, coexisting universes — he delves deeper than most. Many others have written about the multiverse, but Tegmark gives his multiverse four levels, each one infinitely vaster than the last.

The first three of these may already be familiar to physics fans — certainly to “hard-core” fans. Level I suggests a Library of Babel scenario. Tegmark claims that within an infinitely large universe, all worlds compatible with the laws of physics must literally exist, which leads him to observe that, “if there are indeed many copies of ‘you,’ with identical past lives and memories, this kills the traditional notion of determinism.” A questionable contention, but pretty prosaic compared to the madness to come. Level II, an idea from the string theory landscape, proposes that unaffiliated universes may exist to accommodate different forms of the laws of nature, which “explains” why our laws have their particular form by eliding the question entirely. Level III proposes that, in each of these universes, quantum probabilities refer to events that coexist, just more or less often, in the manner of the “many worlds” version of quantum mechanics.

I personally think Levels II and III are unformed, but each has too dense a paper trail to dismiss without effort. But Tegmark isn’t content with just synthesizing old arguments. With Level IV, the logic of these prior unobservables is stretched far past its breaking point.

The Level IV multiverse is an extreme twist on Murphy’s law: it supposes that every mathematical structure that can exist, does exist. In Tegmark’s vision, this collapse of mathematical and physical existence explains not only our universe, but all the rest combined.

Tegmark approaches this conclusion with ever more unsubstantiated assertions. The External Reality Hypothesis (“there is an external physical reality completely independent of us humans”) sounds plausible. This hypothesis paired with general relativity allows us to picture the universe as a four-dimensional geometric structure, which leads to the Mathematical Universe Hypothesis (“a mathematical structure is our external reality, rather than being merely a description thereof”). To sharpen these claims, Tegmark then introduces the Computable Universe Hypothesis (“our external physical reality is defined by computable functions”) and the Finite Universe Hypothesis (“our external physical reality is a finite mathematical structure”).

When the ERH, MUH, CUH, and FUH are all simultaneously in play, the flurry of acronyms distracts from the basic argument. But the part that should arouse suspicion is the Mathematical Universe Hypothesis — the assertion that all mathematical descriptions are physical things.

To his credit, Tegmark doesn’t hide that this seems wrong. In high school geometry, we are taught that perfect circles don’t physically exist, because circles, like points, planes, angles, and all the rest, are abstractions, not directly linked to experience. Tegmark argues otherwise. Tegmark quotes the standard definition (penned by David Hilbert) of mathematical existence — “Mathematical existence is merely freedom from contradiction” — and asserts that mathematical existence implies physical existence. Every mathematical structure, from simple circle to quantum cosmos, is physically real within the Level IV multiverse. Tegmark welcomes them all. The experimentalist’s job, in his view, is just to figure out which one we inhabit.

What’s more, Tegmark is confident that the “MUH is in principle testable and falsifiable,” since we might find nonmathematical physical systems. (How, exactly?) Another benefit: “Exploring the Level IV multiverse doesn’t require rockets or telescopes, merely computers and ideas.”

By the end of Our Mathematical Universe, only Tegmark’s friendly demeanor, marked by Douglas Adams quotes and personal asides, convinced me he wasn’t just trolling. The Level IV multiverse, after all, breaks basic scientific rules. There are no experiments that can test for that which is a priori unobservable. But Tegmark, ever the gentleman, acknowledges all this, even going so far as to share his “Dr. Jekyll/Mr. Hyde Strategy” for balancing mainstream research with batshit diversions. And he still manages to float off on charm.

Let me put it this way: as popular science, I think Our Mathematical Universe is about 70 percent bullshit, but it’s 100 percent entertaining. And as an unreliable narrator, Mad Max is a marvel — a character with a doctrine so absurd that, were his book labeled as fiction, it would read as dry parody, a reductio ad absurdum for the Level II and III multiverses, and a challenge to the disciples. My enjoyment of the book was undercut only when I reflected that it was labeled as “nonfiction” and “science,” and yet flouts the definitions of both words.


Exhibit B: The Curator

“If you read the preceding paragraphs and are not shocked or perplexed then you must go back and read them again. If you skipped them, proceed and be shocked.”

The above injunction, from late in The Island of Knowledge: The Limits of Science and the Search for Meaning, promises another way to zonk our gourds. But Marcelo Gleiser, a professor of natural philosophy, physics, and astronomy at Dartmouth College, doesn’t argue for any new scheme. To quote the jacket copy, Gleiser “reaches a provocative conclusion: science, the main tool we use to find answers, is fundamentally limited. The essence of reality is unknowable.”

But one doesn’t need to go far past the introduction to discover that Gleiser is no Feyerabendian bombardier. The central metaphor for his book imagines knowledge as a small island in a dark ocean of unknowns, but Gleiser’s faith in science is nonetheless secure enough to assert that “science is the best toolkit we have to create a universal language that transcends individual difference.”

Not exactly the words of a madman on a quest — which is the best and worst thing about the book. Gleiser is unvaryingly reasonable, but he won’t shock you. In fact, he might not even hold your attention. At most, you might be mildly perplexed.

Of course, reasonableness has its benefits. Gleiser is adept at summarizing beliefs that aren’t his own, always steering clear of Whig history. When he discusses Ptolemy’s geocentric planetary model, for example, he doesn’t pretend it was inherently ridiculous. He describes why Copernicus disliked Ptolemy’s equant (the point is slightly offset from Earth’s center around which the planets and Sun are all supposed to circle), and tells us how the switch to heliocentrism was inspired not by new data, but by Copernicus’s irritation at this “irregularity.” Once better data was available, Kepler found that the Sun lies on the focus of an ellipse — a reversion to Ptolemaic irregularity and away from Copernican perfection.

Investigating the dark corners of well-known results allows Gleiser, at his best, to challenge science’s possible shortsightedness. In Chapter 11, “Cosmic Blindness,” he remarks that if our current view of the universe, with its increasing outward acceleration, is correct, then in the far future, some two trillion years from now (current age: 13.8 billion years), any observers will see only darkness outside their own galaxies. Their best observations will force them to adopt conclusions quite different from those of today’s science. “Ironically,” Gleiser writes, “their cosmology will return to that of a static cosmos, an island universe comprising the galaxies within the Local Supercluster, surrounded by a vast expanse of dark, empty space.”

In moments like this, I felt a twinge of hope that our friend Marcelo would write something interesting, rather than just dignified. Unfortunately, this isn’t that kind of a book. “I don’t mention [this despairing prospect] as something for us to worry about but to reflect upon,” Gleiser consoles, and by the bottom of the page he’s fully defused his insight. “To avoid the funk of a modern scientific nihilism,” his homily continues, “we must find joy in what we are able to learn of the world, even if knowing that we can be certain of very little.”

What joy! But would it have hurt to venture further than received wisdom?

The main problem is Gleiser’s timidity, which becomes annoying at the end of Part I, when we’re plodding through the multiverse (here, Gleiser just summarizes the opinions of others). It gets worse in Part II, on quantum theory, when evasiveness enables a tacit alignment with the status quo. “To a physicist,” Gleiser claims, “what something ‘is’ is less relevant than whether her explanations are efficient.” But he never defends “efficiency” as a route to validation.

Furthermore, Gleiser writes that “explaining reality may be too much of a tall order, even for science.” But this skepticism constricts his judgment. It’s not that Gleiser doesn’t know his options. It’s just that he’s not interested. He ably discusses Bohmian mechanics, a system invented by David Bohm to make quantum mechanics deterministic. In this theory, particles are pushed around by waves, all while following the statistical predictions of traditional quantum theory. But while his scheme may clarify scientific ontology (an electron, for Bohmians, includes a particle and a wave), Gleiser doesn’t bite. He asserts, “We would like to be able to distinguish between competing theories on the basis of experiments; if different theories give the same experimental results, why not pick the simplest one, that is, traditional quantum mechanics without extra pilot waves?”

Notice how Gleiser conflates simplicity and tradition. In this, Gleiser establishes himself as the anti-Tegmark: a conservator of traditional values who expects to astonish us without pushing the material in any discernible direction. From the quantum fundamentals, here’s Gleiser’s pat moral takeaway: “There are aspects of reality that are permanently beyond our reach.”

This conclusion seems too small and too obvious for so strange and incredible a thing as quantum theory. But in Part III, Gleiser continues this deflationary trend. He opines that math is a human construction. He gives Gödel an upbeat spin (“how refreshing that we are not slaves to formal intellectual process!”). He explains why we probably don’t live in a simulation.

Then he insists that “the quest must go on, always” before closing with a Dylan Thomas pastiche.

None of which is offensive, all of which is fine. The Island of Knowledge is carefully considered and thoughtfully written. But the book is ultimately too polite to pursue its own insights. There’s no way around it: the book is pretty boring.

I wish it weren’t. Gleiser respects evidence far more than Tegmark, but for readers, delusion is often preferable to boredom, and an insane doctrine is often preferable to no doctrine at all. The notion that knowledge is bordered by ubiquitous ignorance seems obvious. If Gleiser insists on preserving pieties instead of burning them, he can’t expect us to follow. Most of us want better stories from science. Gleiser doesn’t even give us better ways to doubt.


Exhibit C: The Legislator

To summarize: Tegmark is too brazen, and Gleiser too bashful. Dialectical convention all but requires David Z. Albert to embody a middle way. Philosophically, I think he does, though After Physics seems destined to reach as few readers as possible, and with maximum difficulty.

Albert is a philosophy professor at Columbia University, albeit one with a doctorate in physics. His views are usually straightforward, but not quite straightforwardly stated. For instance, the chapter “The Difference between the Past and the Future” is delivered as a 40-page dialogue between Jedediah and Huckleberry, characters whose styles exactly match that of David Z. Albert. Unfortunately, Albert’s style sits uneasily between Saul Kripke and Lester Bangs. It’s a short book — 181 pages, index included — that reads long. All of which makes After Physics a tough sell. In the preface, Albert admits, “[These eight essays] have various different sorts of connections to one another, and they have been designed to be read in the order in which they are printed, but they aren’t meant to add up to anything like a single, sustained, cumulative, argument.”

A jazzy-baroque style and no central argument — but After Physics is still valuable for readers seriously interested in scientific metaphysics. Albert writes like a philosopher and thinks like a physicist, parsing the logic of each argument while allowing physics to guide his insights. He doesn’t have any one dogma. In a footnote, Albert calls out the many-minds interpretation of quantum mechanics, an interpretation that he invented, as “bad, silly, tasteless, hopeless, explicitly dualist.” His essays, too, split into two: some build schemes up, others tear them down.

For instance, “The Technique of Significables” is a teardown essay with some buildup along the way. In it, Albert argues that a “scientific account of the world that makes the right predictions, under all physically possible circumstances, about the positions of golf balls” will also have to make correct predictions about pretty much everything else. That’s the buildup. But Albert uses this to show that for the usual versions of quantum theory (and for some unusual ones as well), no faster-than-light nonlocal signals can transmit information — or move golf balls, as it were.

Okay, fine: another book to drain the world of wonder. I can see eyes roll in their sockets. But the surprise here is that by sticking close to standard physics, and by refusing to abandon the logic of pursuit, Albert is able to hint at a revelation as strange as anything in Tegmark, and far stranger than the mysterianism of Gleiser.

Per the opening disclaimer, this argument isn’t presented as a single, sustained, or cumulative whole. Instead, it’s nestled among the quantum analyses that make up the back half of After Physics. (The front half concerns itself mainly with time and thermodynamics — cf. the bit with Jedediah and Huckleberry — which I’m mostly ignoring here.) But even if these analyses don’t drive toward a grand finale, they have a unified approach, and the rigorous application of this approach is what leads Albert to his unusual claims about the structure of the universe.

This approach is notable for its simplicity. Albert examines the bare formalism of physical theories to ask: What does this formalism, taken literally, tell us about the universe? If we refuse to admit any extra assumptions, what is the math trying to say?

Albert is worried about what he calls the “narratability” of different theories — the ability of these theories to tell coherent stories about the world. He argues we should expect our theories to be narratable, but he also shows that, at present, our standard theories really aren’t. Consider Schrödinger’s cat. Many physicists insist that the cat really isn’t dead or alive until a measurement “collapse” occurs. But these collapses take place at the specific points when and where a measurement occurs, which introduces a tension between quantum theory and relativity. In relativistic theories, it’s possible for different observers not to agree on the order of a sequence of events. This means that for processes that involve multiple measurements, the traditional relativistic quantum-mechanical theories — the best-tested theories that current science has to offer — aren’t narratable, and hence aren’t able to tell coherent stories.

Two sorts of fixes are offered for this. To deal with narrative coherence, Albert suggests we may have to give up Einstein’s vision of relativity, with all its geometrical shifts in observer viewpoints, for an older, interactional vision, dating back to Einstein’s early contemporary, Hendrik Lorentz. As Albert puts it, “The production of geometrical appearances, like the production of appearances generally, is obviously, invariably, at bottom a matter of dynamics.”

And to understand the nature of quantum collapse, Albert suggests we examine GRW “dynamical collapse” theories (so-called after Ghirardi, Rimini, and Weber, inventors of the first of this kind). GRW theories differ from others because they posit that collapses happen physically — that the statistical predictions of quantum theory arise because these collapses occur every so often in time, and every so far in space, based on statistical regularities.

So far, one could dismiss this as technical nitpicking. But the assumed primacy of dynamics (i.e., of the relationships among physical entities as the change in time) leads to an unexpected possibility.

For Bohmian and GRW theories (both strong candidates, where narratability and ontological clarity are concerned), Albert stresses that their formalisms don’t include the three spatial dimensions of everyday experience. For a universe of N particles, these theories require 3N dimensions. So a universe with one particle would be described in three dimensions, a universe with two particles would need six dimensions, and a universe like ours, with roughly 1080 particles, would require some 3 x 1080 dimensions! Albert takes this to mean that the world’s three-dimensional appearance may not be an immutable fact of geometry, but an approximate, macroscopic outcome of dynamics — which leads him, in turn, to conclude that

once all this is taken in, the necessity of somehow making sense of our experience within the high-dimensional space in which the wave function undulates begins to feel like a simple and straightforward and flat-footed and ineluctable matter of physics, and all of the earlier hemming and hawing about the metaphysical character of the wave function begins to feel a little beside the point, and the business of artificially inserting a three-dimensional space directly into the foundations of the world seems (I don’t know) unavailing, and empty, and silly.

Maybe this whole thing is silly. But if it is, it’s just one step beyond experiment. And should our universe turn out not to follow Bohmian or GRW rules, Albert’s knotty analyses will continue to be useful for understanding whatever types of theories might come next.



So, after the long buildup, what’s the summary or point or moral of all this?

How about (to be briefly cryptic) a Jesus quote, from the Gospel of Mark: “A prophet is not without honor, except in his hometown and among his relatives in his own household.”

I’ve already opined that David Z. Albert offers a more piercing analysis of modern physics than Marcelo Gleiser, but that Gleiser’s recitations are still more convincing than Max Tegmark’s mathematical abyss. I’ve also opined that this ordering might be flipped for beach readers out for a good time. After Physics is rife with technical language about counterfactuals and density matrices, and will take effort for anyone to read — even physicists. The Island of Knowledge, conversely, isn’t too difficult, but it’s a little too familiar, a little too careful. Our Mathematical Universe hits the sweet spot: unfamiliar but comprehensible, and weird enough to be fun.

But the more I’ve considered this inversion, the more troubling it seems.

The quote about prophets and hometowns came to mind, not because any of these writers are religious (so far as I know), but for the opposite reason — because physicists, of all people, should not be prophets. Competent physicists understand how their explanatory schemata are based on either empirical facts or introduced assumptions. But even if there’s no especial honor among peers, once physicists venture out from hometown they may as well be prophets. And make no mistake: most of us who read popular science want prophets. We want the inside scoop, an easy path to transcendence, oracles who will just give us the truth.

The good news is that there are plenty of difficult, boring books that will do just that. The bad news is that there are just as many simple, exciting books that only want to blow your mind.

Which brings me to my own stab at prophecy. I predict that these quantum ambiguities will someday all be solved, that the conceptual issues of quantum mechanics will leave philosophy to join the solid realm of scientific knowledge. When that happens, Schrödinger’s cat will die quietly, smothered in old books. By then, we’ll wear new confusions on our T-shirts.


David Kordahl lives in Mesa, Arizona, where he teaches science.